Consider a two-dimensional flow \((\tilde{x}, \tilde{y})\) of unsteady magneto-hydrodynamic nanofluid in an asymmetric micro-channel, in which wave propagation is along the *x̃* direction (Fig. 1). This flow is formed by the propagation of a sinusoidal wave at a constant speed *c* along the channel having elastic walls. The combination of externally applied magnetic field, electric field and pressure gradient affects the driving fluid. It is supposed that the electric field \(E_{0}\) is imposed axially, and magnetic field \(B_{0}\) is transversely of the fluid flow. Let \(\tilde{y}_{1} = \tilde{h}_{1} ( \tilde{x}, \tilde{{t}} )\) and \(\tilde{y}_{2} = \tilde{h}_{2} ( \tilde{x}, \tilde{{t}} )\) be the upper and lower walls of channel, respectively:

$$\begin{aligned}& \tilde{h}_{1} ( \tilde{x}, \tilde{{t}} ) = d_{1} + a_{1} \cos ^{2} \biggl( {\frac{ ( \tilde{x} - c \tilde{t} ) \pi}{ \lambda}} \biggr), \end{aligned}$$

(1)

$$\begin{aligned}& \tilde{h}_{2} ( \tilde{x}, \tilde{{t}} ) = - d_{2} - a_{2} \cos ^{2} \biggl( {\frac{ ( \tilde{x} - c \tilde{t} ) \pi}{\lambda}} + \varphi \biggr), \end{aligned}$$

(2)

where \(\tilde{h}_{1} ( \tilde{x}, \tilde{y} )\), \(\tilde{h}_{2} ( \tilde{x}, \tilde{y} )\), \(d_{1}\), \(d_{2}\), \(a_{1}\), \(a_{2}\), *φ*, *λ* and *t̃* are the upper wall, the lower wall, constant height of upper wall measured from \(\tilde{y}_{1} = 0\), constant height of lower wall measured from \(\tilde{y}_{2} = 0\), amplitude of the upper and lower walls, phase difference, wavelength and time, respectively.

### 2.1 Distribution of potential

Ion separation occurs during EOF, and EDL is formed near the channel walls, creating an electric potential *ϕ̃* difference. The Poisson–Boltzmann equation is used to describe the *ϕ̃* in the microchannel:

$$\begin{aligned}& \nabla^{2} \tilde{\phi} =- \frac{\rho_{e}}{\in\in_{0}}, \end{aligned}$$

(3)

where \(\rho_{e}\), ∈, ∈_{0} and *ϕ̃* are the net charge density, the relative permittivity of the medium, the permittivity of free space (\(8.854 \times 10^{- 12}\) F⋅m^{− 1}) and electric potential distribution. The probability of detecting ions at a specific position in electric double layer (EDL) is relative to Boltzmann factor \(e^{( e z_{v} \tilde{\phi} / T_{\mathrm{av}} K_{B} )}\). The positive \(( n^{+} )\) and negative ions \(( n^{-} )\) can be explained by the number density of the Boltzmann equation:

$$ n^{\pm} = n_{0} e^{( \pm \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\phi} )}, $$

(4)

where the average numbers of negative and positive ions are denoted by \(n_{0}\). The distribution of ionic concentration is considered to be effective when there is no ionic concentration gradient in the axial direction of the microchannel. By the electrolyte symmetry assumption, the total charge density \(\rho_{e}\) is taken as

$$\begin{aligned}& \rho_{e} = - z_{v} e \bigl( n^{-}- n^{+} \bigr)= - 2 z_{v} e n_{0} \sinh \biggl( \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\phi} \biggr). \end{aligned}$$

(5)

In the above, \(z_{v}\), *e*, \(T_{\mathrm{av}}\) and \(K_{B}\) are the ions valence, electron charge, average temperature, and Boltzmann constant. The nonlinear terms in the Nernst–Planck equations are \(O( P_{e} \alpha^{2} )\), where \(P_{e} = R_{e} S_{c}\) represents the ionic Peclet number and \(S_{c}\) is the Schmidt number. Assume that the Peclet number is very small.

Now, by means of Eqs. (3)–(5),we approximate Eq. (3) as:

$$\begin{aligned}& \frac{{d}^{{2}} \tilde{\phi}}{{d} \tilde{y}^{{2}}} = \frac{{2} z_{v} e n_{0}}{\in\in_{0}} \sinh \biggl( \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\phi} \biggr). \end{aligned}$$

(6)

The boundary conditions for dimensional form *Φ̃* can be written as

$$\begin{aligned}& \tilde{\phi}= \tilde{\zeta}_{1}\quad \text{at } \tilde{y}_{1} = \tilde{h}_{1} ( \tilde{x}, \tilde{{t}} ), \end{aligned}$$

(7a)

$$\begin{aligned}& \tilde{\phi}= \tilde{\zeta}_{2}\quad \text{at } \tilde{y}_{2} = \tilde{h}_{2} ( \tilde{x}, \tilde{{t}} ), \end{aligned}$$

(7b)

where \(\tilde{\zeta}_{1}\) and \(\tilde{\zeta}_{2}\) are the zeta potentials at the upper and lower walls, respectively. In order to proceed with dimensionless variables, we introduce:

$$\begin{aligned}& \begin{aligned} &a= \frac{d_{2}}{d_{1}},\qquad b= \frac{a_{1}}{d_{1}}, \qquad c= \frac{a_{2}}{ d_{2}},\qquad h_{1 }= \frac{\tilde{h}_{1}}{d_{1}},\qquad h_{2} = \frac{\tilde{h}_{2}}{d_{1}}, \\ & m= \frac{d_{1}}{{\lambda D}},\qquad p= \frac{\tilde{{p}} d_{1}^{{ 2}}}{c_{\lambda} \mu_{f}},\qquad t= \frac{c \tilde{{t}}}{{\lambda}},\qquad u= \frac{\tilde{u}}{c},\qquad v= \frac{\tilde{v}}{c \alpha}, \\ &x= \frac{\tilde{x}}{{\lambda}},\qquad y= \frac{\tilde{y}}{d_{1}},\qquad B_{r} = E_{c}\cdot {P_{r}},\qquad E_{c} = \frac{c^{{ 2}}}{c_{p} ({T_{1} - T _{0}} )}, \\ & H_{r} = B_{0} d_{1} \sqrt{ \frac{{\sigma_{e}}}{\mu_{f}}},\qquad N_{b} = \frac{{\gamma 1} ({C_{1} - C_{0}} ) D_{B}}{\nu_{f}}, \\ &{N_{t}} = \frac{{\gamma_{1}} ({T_{1} - T_{0}} ) D_{T}}{{T_{m}} \nu_{f}},\qquad {P_{r}}= \frac{\mu_{f} c_{p}}{ k_{f}},\qquad R_{e} = \frac{\rho_{f} c d_{1}}{\mu_{f}}, \\ & S_{c} = \frac{{c} d_{1}}{K_{B}},\qquad U_{\mathrm{HS}} =- \frac{E_{0} \in\in_{0} T_{\mathrm{av}} K_{B}}{e z_{v} \mu_{f}},\qquad \alpha= \frac{d_{1}}{\lambda}, \\ &\beta= \frac{U_{\mathrm{HS}}}{c},\qquad \gamma_{1} = \frac{ ( \rho c )_{p}}{ ( \rho c )_{f}}, \qquad \gamma_{2} = \frac{{\sigma_{e}} d_{1}^{2} E_{0}^{2}}{{k} ({T_{1} - T_{0}} )},\qquad {\gamma_{3}} =P_{r} \gamma_{2}, \\ & \nu_{f} = \frac{\mu_{f}}{\rho_{f}},\qquad \lambda_{D} = \frac{1}{e z_{v}} \sqrt{\frac{T_{\mathrm{av}} K_{B} \in\in_{0}}{{2} n_{0}}},\qquad \zeta_{1} = \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\zeta}_{1}, \\ &\zeta_{2} = \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\zeta_{2}}, \qquad \phi= \frac{e z_{v}}{T_{\mathrm{av}} K_{B}} \tilde{\phi}, \\ & \varTheta= \frac{\tilde{{T}}- T_{0}}{{T_{1} - T_{0}}},\qquad \varOmega= \frac{{C- C_{0}}}{ {C_{1} - C_{0}}}. \end{aligned} \end{aligned}$$

(8)

By using the dimensionless variables defined in Eq. (8), Eqs. (6) and (7a)–(7b) become

$$\begin{aligned}& \frac{{d}^{{2}} \phi}{{d} y^{{2}}} = \phi m^{2}, \end{aligned}$$

(9)

$$\begin{aligned}& \begin{aligned} &\phi= \zeta_{1} \quad \text{at } y_{1} = h_{1} ( x,t ), \\ &\phi= \zeta_{2} \quad \text{at } y_{2} = h_{2} ( x,t ). \end{aligned} \end{aligned}$$

(10)

Moreover, we suppose that the zeta potential at walls is small enough that the Debye–Hückel linearization is approximately applicable. The linear Poisson–Boltzmann equation is solved by using the boundary conditions given in Eq. (10) to obtain the function of potential distribution

$$\begin{aligned} \phi =& \biggl( \frac{\zeta_{2} \sinh ( m h_{1} ) - \zeta_{1} \sinh ( m h_{2} )}{\sinh ( m h_{1} -m h_{2} )} \biggr) \cosh ( my ) \\ &{} + \biggl( \frac{\zeta_{1} \cosh ( m h_{2} ) - \zeta_{2} \cosh ( m h_{1} )}{\sinh ( m h_{1} -m h_{2} )} \biggr) \sinh ( my ). \end{aligned}$$

(11)

Here, *m* is the electroosmotic parameter. If we put \(\zeta_{1} = \zeta_{2}\), then the solution of Eq. (11) reduces to the results of [8].

### 2.2 Analysis of flow

Taking into account the viscous dissipation and Joule heating effects, the governing equations for electroosmotically conducting nanofluid affected by the peristaltic flow in asymmetric microchannel are expressed here as:

$$\begin{aligned}& \frac{\partial \tilde{u}}{\partial \tilde{x}} + \frac{\partial \tilde{v}}{ \partial \tilde{y}} =0, \end{aligned}$$

(12)

$$\begin{aligned}& \rho_{f} \biggl( \frac{\partial \tilde{u}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{u}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{u}}{\partial \tilde{y}} \biggr) = - \frac{\partial \tilde{p}}{ \partial \tilde{x}} + \mu_{f} \biggl( \frac{\partial^{2} \tilde{u}}{ \partial \tilde{x}^{2}} + \frac{\partial^{2} \tilde{u}}{\partial \tilde{y}^{2}} \biggr) + \rho_{e} E_{0} -\sigma_{e} B_{0}^{2} \tilde{u}, \end{aligned}$$

(13)

$$\begin{aligned}& \rho_{f} \biggl( \frac{\partial \tilde{v}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{v}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{v}}{\partial \tilde{y}} \biggr) =- \frac{\partial \tilde{p}}{\partial \tilde{y}} + \mu_{f} \biggl( \frac{\partial^{2} \tilde{v}}{\partial \tilde{x}^{2}} + \frac{\partial^{2} \tilde{v}}{\partial \tilde{y}^{2}} \biggr), \end{aligned}$$

(14)

$$\begin{aligned}& \biggl( \frac{\partial \tilde{T}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{T}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{T}}{\partial \tilde{y}} \biggr) = \frac{ k_{f}}{ (\rho c)_{f}} \biggl( \frac{\partial^{2} \tilde{{T}}}{\partial \tilde{x}^{2}} + \frac{\partial^{2} \tilde{T}}{\partial \tilde{y}^{2}} \biggr) \\& \hphantom{ \bigg( \frac{\partial \tilde{T}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{T}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{T}}{\partial \tilde{y}} \bigg) =} {} +\gamma_{1} \biggl[ D_{B} \biggl( \frac{\partial \tilde{T}}{ \partial \tilde{x}} \frac{\partial \tilde{C}}{\partial \tilde{x}} + \frac{\partial \tilde{T}}{\partial \tilde{y}} \frac{\partial \tilde{C}}{\partial \tilde{y}} \biggr) + \frac{D_{T}}{T_{m}} \biggl\{ \biggl( \frac{\partial \tilde{T}}{\partial \tilde{x}} \biggr)^{2} + \biggl( \frac{\partial \tilde{T}}{\partial \tilde{y}} \biggr)^{2} \biggr\} \biggr] \\& \hphantom{ \bigg( \frac{\partial \tilde{T}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{T}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{T}}{\partial \tilde{y}} \bigg) =} {}+ \frac{\varPhi}{(\rho c)_{f}} + \frac{{\sigma_{e}} B_{0}^{2} \tilde{u}^{2}}{(\rho c)_{f}} + \frac{{\sigma_{e}} E_{0}^{2}}{(\rho c)_{f}}. \end{aligned}$$

(15)

Here *Φ* represents the viscous dissipation and is mathematically expressed as:

$$\begin{aligned}& \begin{aligned}&\varPhi = \mu_{f} \biggl[ 2 \biggl( \frac{\partial \tilde{u}}{\partial \tilde{x}} \biggr)^{2} +2 \biggl( \frac{\partial \tilde{v}}{\partial \tilde{y}} \biggr)^{2} + \biggl( \frac{\partial \tilde{u}}{\partial \tilde{y}} + \frac{\partial \tilde{v}}{\partial \tilde{x}} \biggr)^{2} \biggr], \\ & \biggl( \frac{\partial \tilde{C}}{\partial \tilde{t}} + \tilde{u} \frac{\partial \tilde{C}}{\partial \tilde{x}} + \tilde{v} \frac{\partial \tilde{C}}{\partial \tilde{y}} \biggr) = D_{B} \biggl( \frac{\partial^{2} \tilde{C}}{\partial \tilde{x}^{2}} + \frac{\partial^{2} \tilde{C}}{\partial \tilde{y}^{2}} \biggr) + \frac{D_{T}}{T_{m}} \biggl( \frac{\partial^{2} \tilde{T}}{\partial \tilde{x}^{2}} + \frac{\partial^{2} \tilde{T}}{\partial \tilde{y}^{2}} \biggr). \end{aligned} \end{aligned}$$

(16)

Here \(( \tilde{u}, \tilde{v} )\) are the components of velocity along the *x̃* and *ỹ* direction, respectively. Also, \(\rho_{f}\), \(\mu_{f}\), \(\sigma_{e}\), *p̃*, *T̃*, \(k_{f}\), \((\rho c)_{f}\), *C̃*, \(\gamma_{1}\), \(D_{B}\) and \(D_{T}\) represent the density of the fluid, dynamic viscosity of the fluid, electrical conductivity, pressure field, temperature, thermal conductivity of the fluid, heat capacity of the fluid, concentration field, ratio of the effective heat capacity of the nanoparticle to the heat capacity of the fluid, coefficient of thermophoresis diffusion, and coefficient of Brownian motion, respectively. The terms appearing on the left-hand side of Eq. (13) are inertial forces (due to the convection or bulk motion) and the first term on the right-hand side is because of pressure gradient, while the second and third terms are due to viscosity or advection, the fourth term is because of electrical force per unit volume, and the last term is due to magnetic body (per unit volume) forces. Furthermore, the last three terms appearing on the right-hand side of Eq. (15) represent dissipation due to friction, magnetic and electric field, respectively.

Using Eq. (8), the non-dimensional variables, in Eqs. (13)–(16), Eq. (12) is satisfied and Eqs. (13)–(16) become

$$\begin{aligned}& R_{e} \alpha \biggl( \frac{\partial}{\partial t} +{u} \frac{\partial}{ \partial x} + v \frac{\partial}{\partial y} \biggr)u = - \frac{\partial p}{\partial x} + \biggl( \alpha^{2} \frac{\partial^{2}}{ \partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \biggr) u + \beta \phi m^{2}- H_{r}^{2}u, \end{aligned}$$

(17)

$$\begin{aligned}& R_{e} \alpha^{3} \biggl( \frac{\partial}{\partial t} +u \frac{\partial}{ \partial x} +v \frac{\partial}{\partial y} \biggr) v=- \frac{\partial p}{ \partial y} + \alpha^{2} \biggl( \alpha^{2} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \biggr) v, \end{aligned}$$

(18)

$$\begin{aligned}& R_{e} \alpha \biggl( \frac{\partial}{\partial t} +_{u} \frac{\partial}{ \partial x} +v \frac{\partial}{\partial y} \biggr) \varTheta \\& \quad = \frac{1}{ P_{r}} \biggl( \alpha^{2} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \biggr) \varTheta + E_{c} \biggl[ 2 \alpha^{2} \biggl( \frac{\partial_{u}}{\partial x} \biggr)^{2} +2 \alpha^{2} \biggl( \frac{\partial v}{\partial y} \biggr)^{2} + \biggl( \frac{\partial_{u}}{\partial y} + \alpha^{2} \frac{\partial v}{\partial x} \biggr)^{2} \biggr] \\& \qquad {} + \biggl[N_{b} \biggl( \alpha^{2} \frac{\partial \varOmega}{\partial x} \frac{\partial \varTheta}{\partial x} + \frac{\partial \varOmega}{\partial y} \frac{\partial \varTheta}{\partial y} \biggr) +{N_{t}} \biggl\{ \alpha^{2} \biggl( \frac{\partial \varTheta}{\partial x} \biggr)^{2} + \biggl( \frac{\partial \varTheta}{\partial y} \biggr)^{2} \biggr\} \biggr] \\& \qquad {} +{\gamma_{2}} + E_{c} H_{r}^{2}u , \end{aligned}$$

(19)

$$\begin{aligned}& \alpha \biggl( \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} \biggr) \varOmega = \frac{1}{S_{c}} \biggl( \alpha^{2} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \biggr) \varOmega + \frac{1}{S_{c}} \frac{{N_{t}}}{{N_{b}}} \biggl( \alpha^{2} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \biggr) \varTheta. \end{aligned}$$

(20)

Here, \(R_{e}\), *α*, *β*, \(H_{r}\), \(P_{r}\), \(E_{c}\), \(N_{b}\), \(N_{t}\), \(S_{c}\), *Θ* and *Ω* are the Reynolds number, wave number, mobility of the medium, Hartmann number, Prandtl number, Eckert number, thermophoresis parameter, Brownian motion parameter, Schmidt number, dimensionless temperature and concentration field, respectively.

Applying a long-wavelength approximation, ignoring the term with a high power of *α*, Eqs. (17)–(20) reduce to

$$\begin{aligned}& \frac{\partial p}{\partial x} = \frac{\partial^{2} u}{\partial y^{2}} + \beta \phi m^{2}- H_{r}^{2}u, \end{aligned}$$

(21)

$$\begin{aligned}& \frac{\partial p}{\partial y} =0, \end{aligned}$$

(22)

$$\begin{aligned}& \frac{\partial^{2} \varTheta}{\partial y^{2}} + B_{r} \biggl( \frac{\partial_{u}}{\partial y} \biggr)^{2} + P_{r} N_{b} \biggl( \frac{\partial \varOmega}{\partial y} \frac{\partial \varTheta}{\partial y} \biggr) + P_{r} N_{t} \biggl( \frac{\partial \varTheta}{\partial y} \biggr)^{2} +\gamma_{3} + B_{r} H_{r}^{2}u = 0, \end{aligned}$$

(23)

$$\begin{aligned}& \frac{1}{S_{c}} \frac{\partial^{2} \varOmega}{\partial y^{2}} + \frac{1}{ S_{c}} \frac{{N_{t}}}{{N_{b}}} \frac{\partial^{2} \varTheta}{\partial y^{2}} + N_{t} \biggl( \frac{\partial \varTheta}{\partial y} \biggr)^{2} = 0. \end{aligned}$$

(24)

By using cross-differentiation, we have eliminated the pressure term from the dimensionless Eqs. (17) and (18), and can write it as a single nonlinear differential equation. Now let us define *Ψ*, the stream function, as \({u} = \frac{\partial \varPsi}{\partial y}\), \(v = - \frac{\partial \varPsi}{\partial x}\), satisfying the continuity Eq. (10). Equations (21), (23) and (24) can be expressed as a stream function using:

$$\begin{aligned}& \frac{\partial p}{\partial x} = \frac{\partial^{3} \varPsi}{\partial y^{3}} + \beta \phi m^{2}- H_{r}^{2} \frac{\partial \varPsi}{\partial y}, \end{aligned}$$

(25)

$$\begin{aligned}& \frac{\partial^{4} \varPsi}{\partial y^{4}} - H_{r}^{2} \frac{\partial^{2} \varPsi}{\partial y^{2}} + \beta m^{2} \frac{\partial \varPhi}{\partial y} =0, \end{aligned}$$

(26)

$$\begin{aligned}& \frac{\partial^{2} \varTheta}{\partial y^{2}} + P_{r} N_{b} \biggl( \frac{\partial \varOmega}{\partial y} \frac{\partial \varTheta}{\partial y} \biggr) + P_{r} N_{t} \biggl( \frac{\partial \varTheta}{\partial y} \biggr)^{2} +{\gamma_{3}} + B_{r} \biggl( \frac{\partial^{2} \varPsi}{\partial y^{2}} \biggr)^{2} + H_{r}^{2} B_{r} \biggl( \frac{\partial \varPsi}{\partial y} \biggr)^{2} =0, \end{aligned}$$

(27)

$$\begin{aligned}& \frac{\partial^{2} \varOmega}{\partial y^{2}} + \frac{{N_{t}}}{{N_{b}}} \frac{\partial^{2} \varTheta}{\partial y^{2}} =0. \end{aligned}$$

(28)

The boundary conditions with *Ψ* as a stream function are:

$$\begin{aligned}& \frac{\partial \varPsi}{\partial y} =0,\qquad \varPsi = \frac{F}{2},\qquad \varTheta =0, \qquad \varOmega =0 \quad \text{at } y= h_{1} ( x,t ), \end{aligned}$$

(29a)

$$\begin{aligned}& \frac{\partial \varPsi}{\partial y} =0,\qquad \varPsi =- \frac{F}{2}, \qquad \varTheta =1,\qquad \varOmega =1 \quad \text{at } y= h_{2} ( x,t ). \end{aligned}$$

(29b)

Here the no slip conditions are imposed at the walls of the channel. Also, we have introduced two extra stream function boundary conditions for the purpose of solving a fourth degree differential equation. The flow rate *F* in its non-dimensional form is defined as \(F = A_{0} e^{-Bt}\), where *B* and \(A_{0}\) are constants. The negative or positive flow rates are dependent on the value of constant \(A_{0}\). If \(A_{0} <0\) then \(F <0\); similarly, \(F >0\) if \(A_{0} >0\). A positive flow rate indicates that the flow is in the direction of peristaltic pumping. Negative flow refers to the opposite of flow and peristaltic motion, also known as reverse pumping. It was experimentally found in [9] that the blood flow rate decreases exponentially with the passage of time. The authors of that paper also depicted that the deviation of blood flow rate is independent on the structural aspects of the microchannel.